By V. I. Danilov (auth.), I. R. Shafarevich (eds.)
This EMS quantity involves components. the 1st half is dedicated to the exposition of the cohomology thought of algebraic kinds. the second one half bargains with algebraic surfaces. The authors, who're famous specialists within the box, have taken pains to offer the cloth conscientiously and coherently. The publication comprises a number of examples and insights on quite a few themes. This ebook can be immensely valuable to mathematicians and graduate scholars operating in algebraic geometry, mathematics algebraic geometry, advanced research and similar fields.
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Extra info for Algebraic Geometry II: Cohomology of Algebraic Varieties. Algebraic Surfaces
Those deformations are closely related to cohomology of the tangent sheaf of X (Palamodov (1986)). Example 3. ~) provide important invariants of the variety. }c) or H 1 (0x ). 2. Serre's Theorem. The following theorem established in (Serre (1955)) is a corner-stone in the cohomology theory of coherent sheaves. It is an algebraic analog of Cartan's theorem (Theorem B). Theorem. Let F be a quasi-coherent sheaf on an affine va'riety X. Then Hq(X, F) = 0 for all q > 0. One can show that converse is also true in the Noetherian case: if the cohomology of every quasi-coherent sheaf on a scheme X are trivial, then X is an affine scheme (Hartshorne (1977)).
Theorem (Grothendieek). For every a E K(X) eh(fka) ·td(Ty) = f*(eh(a) ·td(Tx)). 6. Principle of the Proof. As it often happens, it is easier to prove a general formula than its special case - the formula helps itself. In the case in question, we are dealing with a morphism instead of a fixed variety. Clearly, if Grothendieek's formula is valid for morphisms f: X ---+ Y and g: Y ---+ Z, then it also valid for the eomposition g o f: X ---+ Z. This follows essentially from the Leray spectral sequenee, which gives (g o f)k = 9k o fk.
Let E be a locally free sheaf of rank r on a smooth algebraic variety X. We consider the corresponding projective bundle 1r: IP'(E) -+ X, and the tautological invertible sheaf 0(1) on IP'(E). Let ~ denote the divisordass in A 1 (1P'(E)) corresponding to CJ(l). It is known that A(IP'(E)) is freely generatedas an A(X)-module by l,~, ... ,~r-l (Danilov (1988), Chap. 3). So, we get the following expression for ~r in termsofthat basis: In this decomposition, the coefficients ci = ci(E) are said tobe Chern classes of the sheaf E.